One of the more processing intensive problems related to electronic circuit design is that of determining the electromagnetic coupling of a circuit. This function is typically carried out by doing an iterative evaluation of the circuit basis functions. The electromagnetic coupling for a multiple layer integrated circuit (IC) depends upon both the topology and geometry of the circuit. The topology is generally defined by data indicating the layout of electronic components on one or more layers of an insulating substrate, for example, by a schematic diagram that shows how the components are electrically interconnected, but not their physical disposition. By comparison, the geometry of a circuit specifies the physical layout of the circuit and is often indicated by a plurality of data files, such as Graphic Data System II (GDSII) and CalTech Intermediate Format (CIF) files. GDSII files are in a binary format that hierarchically represents planar geometric shapes, text labels, and other information about the layout of integrated circuits (ICs), while CIF files include machine/human readable textual hierarchical commands that define layers and describe the geometry of rectangles, traces, and pads found on IC masks used to create the circuit.
Boundary element method (BEM)-based field solvers are becoming popular for analyzing distributed field behavior in microelectronic circuits. To simulate large-scale microelectronic structures using BEM, a fast iterative solution is extremely useful to overcome time and memory bottlenecks posed by the dense matrices involved in a BEM formulation. However, convergence is a problem with an iterative solution of real-life microelectronic structures having very closely-spaced thin metal layers, due to the poor spectral properties of the system matrix. Therefore, a good linear complexity pre-conditioner is mandatory to improve the spectral properties of the BEM system matrix and to obtain a solution in a reasonable number of iterations using, for example, a Krylov subspace-based iterative solver.
The reasons for poor spectral properties of the BEM matrix are mainly attributed to: (a) decomposition of the divergence-free and divergent components of the currents at lower frequencies; (b) the use of long, thin triangles in the mesh over the geometric description; and, (c) the use of closely-spaced basis functions in the geometry of the BEM matrix. It would be desirable to separately address these two issues in the design and implementation of a three-stage pre-conditioner. For example, item (a) might be addressed by initially carrying out two stages, namely a “Loop-Tree Decomposition” and a “Basis Function Rearrangement.” Problem (c) might be resolved by a third stage, using “Thresholded Incomplete LU Decomposition” (where “LU” refers to the product of a lower triangular matrix and an upper triangular matrix).
For small electrical circuits, the magnetic vector potential (due to solenoidal current) and electric scalar potential contributions (due to divergent current) in electric field integral equation (EFIE) becomes decoupled. Linear Rao-Wilton-Glisson (RWG) basis functions that are traditionally used for modeling EFIE inherently couple the divergence-free and divergent components of current, so at low frequency, the EFIE matrix suffers from the classically known ill-conditioning problem that gives rise to poor convergence properties when an iterative solution is used. The conventional way to solve this problem is by constructing a set of divergence-free (loop) and divergent (loop free—tree) basis functions to model the current. A more effective approach would be to construct loop-tree basis functions for geometries with arbitrary numbers of holes and handles. However, for closely packed, thin three-dimensional (3-D) structures and for non-uniform discretization, even with loop-tree decomposition, the convergence of iterative solver may be poor. An incomplete-LU based pre-conditioner, if used in connection with loop-tree decomposition, should further improve the convergence behavior.
In addition, it would be helpful to employ an efficient technique for creating a 3-D mesh layout for a circuit from a two-dimensional (2-D) layout of the circuit. Such a method for efficiently creating a 3-D mesh layout without the complexity normally encountered should improve the efficiency of the overall solution of the EM coupling problem.